Bi-tilability

Heptiamonds and tetrahexes

It is well known that in the same way the sets of heptiamonds and tetrahexes have the same are -- 168 triangles (= 7 x 24, just as much as the hours in a week). On his website, David Goodger observes that many heptiamond shapes fitting a hexagonal grid are also tetrahex shapes. He provides a nice set of heptiamond shapes that fit a hexagonal grid but he doesn't explicitly specify the ones that can be solved with tetrahexes too.

The difference with the dodecahex problem is that the solutions here are much more numerous and hard to find, or at least to prove, because of the size of the heptiamond set. Here is a beautiful solution with good symmetry (the tetrahex shape was provided by George Sicherman; I discovered the heptiamond tiling):

Light blue regions in the heptiamond shape indicate pieces that can be mirrored together to thus yield more solutions (and actually all solutions for this particular shape).

Because each tetrahex corresponds to 24 triangles, it is not possible that boundaries of individual tetrahexes coincide with boundaries of a subregion of the heptiamond solution. This means that it is not possible to generate further bi-tilable shapes by simply moving around a few of the tetrahexes.

On the other hand, six heptiamonds correspond to 42 triangles, which is enough to complete a figure of seven hexes, six triangles each. In a similar fashion, regions of 12 or 18 heptiamonds can be formed.

If these regions are moved around, we obtain further shapes which may be tilable with the tetrahexes. For instance, the 6 + 18 configuration above proves that the following tetrahex shapes are bi-tilable:

Unfortunately, it is not possible to tile simultaneously 4 heptahexes with the heptiamonds. If this was possible, all the combination of these heptahexes could have been checked for tilability with the tetrahexes and we would have obtained a good amount of bi-tilable shapes. Searching for simultaneously tilable tetrakaidecahexes is more complicated because it is not possible to just try each of the 15,796,897 tetrahexes. I have investigated these questions a little bit more in depth here.

Since there are so many shapes bi-tilable with heptiamonds and tetrahexes, it will probably be impossible to compile an exhaustive list. However, we can search for bi-tilable shapes with special properties. Such are farms: shapes with the biggest possible inner hole. The best I've found so far has a hole of 18 hexagons:

The perimeter of the above farm is 84 (52 for the outer edge and 32 for the inner one). However, George Sicherman has found a better solution with a hole of 19 hexagons and its perimeter is 86. I don't know of any bi-tilable shape with a greater perimeter.

We might also look for solutions with maximum holes. Although tetrahex shapes with seven holes exist, no tiling with heptiamonds for one of them has been found.

There are plenty of six-hole bi-tilable shapes. However, many six hole tetrahexes shapes are untilable with heptiamods because their perimer is high, each hole raising it with 6. This is maybe the reason while no 7-hole shapes have been found and probably don't exist at all.

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